Physics 249 Lecture 27, Nov 7th 2012
Reading: Chapter 9
HW 8 available on web site, due Nov 9th
Exam comments. The exam will cover section 6.6 (transition and reflection of waves),
Chapter 7, Chapter 9 through molecular spectrum, and related material from the HW and
lecture. The exam will again be a mix of challenging questions, interpretation of the
answers to those questions, and simpler computation questions.
1) Covalent bonding
You expect a bound state could exist in cases where there are orbitals with non identical
quantum numbers. As you bring two atoms together there will be a region in between the
atoms where orbitals overlap and there is a higher negative charge density. This region
will attract both protons lowering the overall potential. At very small separations the area
will be smaller leading to a lower total negative charge. For a small enough separation
distance the attractive force will be less than the direct proton repulsion forming the other
side of the potential well. There is also the possibility of exclusion principle repulsion
due to inner orbitals with identical quantum numbers. Proton repulsion can be a more
important since the outer orbitals can overlap if they do not have identical quantum
numbers lowering the effective Z.
It is useful to analyze the system by considering the fact that the electrons are identical
and there is an exchange symmetry, which in the case of the electron, a fermion, requires
that the wave function be anti-symmetric. The wave function is composed of the space
components and the spin component one (only one) of which must be anti-symmetric to
obey they fermion exchange symmetry. Possible wave functions are
For identical particles in 3D central potentials exchanging the particles means essentially
changing their coordinates. Then wave functions that obey the exchange symmetry are
1
𝜓𝑠 = ( ) (𝜓100 (𝑟1 ) + 𝜓100 (𝑟2 ))
√2
𝜓𝐴 = (
1
) (𝜓100 (𝑟1 ) − 𝜓100 (𝑟2 ))
√2
and for spins, introducing 𝜒, for the spin wave function
𝜒𝑠 = (𝜒1 + 𝜒2 )
𝜒𝐴 = (𝜒1 − 𝜒2 )
For an anti symmetric wave function as required for fermions obeying the identical
particle exchange symmetry.
1
𝜓𝐴 = ( ) (𝜓100 (𝑟1 ) − 𝜓100 (𝑟2 ))(𝜒1 + 𝜒2 )
2
or
1
𝜓𝐴 = ( ) (𝜓100 (𝑟1 ) + 𝜓100 (𝑟2 ))(𝜒1 − 𝜒2 )
2
The spins are not dependent on coordinates. The first wave function must be zero at
r1=r2=0. This means the wave function must cross zero and have zero probability at
zero. This prevents bonding since there is no significant charge to attract the protons and
help them overcome their repulsion.
The second solution is the solution we observe. The second solution should have
substantial probability at r=0 since they add. Also since the spin state has the negative
sign the spins must be different or the wave function will be zero everywhere. See
diagram.
2) Covalent and ionic bonds
Ionic bond typically occur between atoms in the far left (low ionization energies) and
atoms on the far right (high electron affinities)(not counting the far right nobel gas
column which has full shells and will not absorb electrons). Ionic bonding can take place
between atoms in rows further from the extreme left and right if the bonding is still
energetically allowed. For instance, in H2O the oxygen atom with two open orbitals in
it’s p shell absorbs (partially) two donated electrons one from each of the two hydrogen
atoms.
Covalent bonds can form between any two (or more) atoms that can have electrons in
orbitals with non identical quantum numbers in the ground state. For instance the
hydrogen atoms in H2 bond covalently. He cannot bond covalently since the 1s2 orbitals
of the both Helium atoms are both full so that the quantum numbers are all identical.
In an ionic bond the electron is typically partially shared rather than fully exchanged
making the bond ionic and part covalent since there are non overlapping orbitals.
Identical atoms that can bond in diatomic molecules will have completely covenant bonds
since neither atoms will see a potential advantage relative to the other atom if it absorbs
the electron.
A purely ionic bond would have a dipole moment: 𝑝𝑖𝑜𝑛𝑖𝑐 = 𝑒𝑟𝑜 . No ionically bonded
molecules are purely ionic. We quantify the fraction that the bond is ionic as the fraction
of the measured dipole moment over the expected dipole moment of an ionic molecule
with the observed separation distance.
3) Other bonds.
Polar molecules
Molecules with dipole bond and align and attract other dipoles. The electron field and
potential energy (𝑈 = −𝑝2 ∙ 𝐸𝑑 ) go as 1⁄𝑟 3 and the force (𝐹 = −𝜕𝑈⁄𝜕𝑟) as 1⁄𝑟 4, which
can be compared to 1⁄𝑟 2 of the coulomb attraction in directly ironically or covalently
bonded molecules.
Nonpolar molecules.
Non polar covalently bonded molecules can be polarized and then form (non)“polar”
molecules. The polarization process gives a dipole momentum of 𝑝2 = 𝛼𝐸𝑑 goes as the
electric field 1⁄𝑟 3 and thus electric potential energy as 𝑈 = −𝑝2 ∙ 𝐸𝑑 = −𝛼𝐸𝑑 ∙ 𝐸𝑑 →
1⁄ and as the force as 1⁄ 7 .
𝑟
𝑟6
Both types of bonds and substantially weaker than ionic or covalent bonds.